Using change of measure arguments that date back at least to [1]}, we reduce the lower bound studying the value of a zero-sum game between an experimenter and nature. This game generalizes a two player game that appears in [1]}, [3]}, and [4]} to treat problems with contexts. In this game, both players suspect the true state is \(\theta =\theta _0\)  and so the best arm is \(I^*(\theta _0)\) . Trying to collect the evidence to certify this, the experimenter picks a collection of positive semi-definite matrices \(M=(M_1, \cdots , M_k)\)  which obey the constraint \(M_1 + \cdots M_k \preceq \mathbb {E}[X_1X_1^\top ]\) . Nature responds by picking an alternative state of nature \(\theta _1\)  that would induce a different optimal arm. Nature's goal is to challenge the experimenter's claim by picking \(\theta _1\)  that makes a certain KL divergence small.
